Abstract: We show that, on a -manifold endowed with a -structure induced by an almost-complex structure, a self-dual (positive) spinor field is the same as a bundle morphism acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of on tangent vectors, and that the squaring map acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.